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Group cohomology, harmonic functions and the first L

Auteur(s)
Bekka, Bachir
Valette, Alain 
Institut de mathématiques 
Date de parution
1997
In
Potential Analysis
Vol.
4
No
6
De la page
313
A la page
326
Mots-clés
  • group cohomology
  • ends
  • harmonic functions
  • L-2-cohomology
  • PROPERTY-T
  • UNITARY REPRESENTATIONS
  • FUNDAMENTAL GROUP
  • KAHLER
  • MANIFOLD
  • INFINITE-GRAPHS
  • RANDOM-WALKS
  • LIE-GROUPS
  • L2-COHOMOLOGY
  • INVARIANT
  • THEOREM
  • group cohomology

  • ends

  • harmonic functions

  • L-2-cohomology

  • PROPERTY-T

  • UNITARY REPRESENTATIO...

  • FUNDAMENTAL GROUP

  • KAHLER

  • MANIFOLD

  • INFINITE-GRAPHS

  • RANDOM-WALKS

  • LIE-GROUPS

  • L2-COHOMOLOGY

  • INVARIANT

  • THEOREM

Résumé
For an infinite, finitely generated group Gamma, we study the first cohomology group H-1(Gamma, lambda(Gamma)) with coefficients in the left regular representation lambda(Gamma) of Gamma on l(2)(Gamma). We first prove that H-1(Gamma, C Gamma) embeds into H-1(Gamma, lambda(Gamma)); as a consequence, if H-1(Gamma, lambda(Gamma)) = 0, then Gamma is not amenable with one end. For a Cayley graph X of Gamma, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Gamma is not amenable, then there is a natural isomorphism between H-1(Gamma, lambda(Gamma)) and HD(X)/C (the latter space being isomorphic to the first L-2-cohomology space of Gamma). We draw the following consequences: (1) If Gamma has infinitely many ends, then HD(X) not equal C; (2) If Gamma has Kazhdan's property (T), then HD(X) = C; (3) The property H-1(Gamma, lambda(Gamma)) = 0 is a quasi-isometry invariant; (4) Either H-1(Gamma, lambda(Gamma)) = 0 or H-1(Gamma, lambda(Gamma)) is infinite-dimensional; (5) If Gamma = Gamma(1) x Gamma(2) with Gamma(1) non-amenable and Gamma(2) infinite, then H-1(Gamma, lambda(Gamma)) = 0.
Identifiants
https://libra.unine.ch/handle/123456789/13868
Type de publication
journal article
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